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<div class="titlepage"><div><div><h3 class="title">
<a name="geometry.spatial_indexes.introduction"></a><a class="link" href="introduction.html" title="Introduction">Introduction</a>
</h3></div></div></div>
<p>
        The Boost.Geometry.Index is intended to gather data structures called spatial
        indexes which may be used to accelerate searching for objects in space. In
        general, spatial indexes stores geometric objects' representations and allows
        searching for objects occupying some space or close to some point in space.
      </p>
<p>
        Currently, only one spatial index is implemented - R-tree.
      </p>
<h5>
<a name="geometry.spatial_indexes.introduction.h0"></a>
        <span class="phrase"><a name="geometry.spatial_indexes.introduction.r_tree"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.r_tree">R-tree</a>
      </h5>
<p>
        R-tree is a tree data structure used for spatial searching. It was proposed
        by Antonin Guttman in 1984 <a href="#ftn.geometry.spatial_indexes.introduction.f0" class="footnote" name="geometry.spatial_indexes.introduction.f0"><sup class="footnote">[1]</sup></a> as an expansion of B-tree for multi-dimensional data. It may
        be used to store points or volumetric data in order to perform a spatial
        query. This query may for example return objects that are inside some area
        or are close to some point in space <a href="#ftn.geometry.spatial_indexes.introduction.f1" class="footnote" name="geometry.spatial_indexes.introduction.f1"><sup class="footnote">[2]</sup></a>. It's possible to insert new objects or to remove the ones already
        stored.
      </p>
<p>
        The R-tree structure is presented on the image below. Each R-tree's node
        store a box describing the space occupied by its children nodes. At the bottom
        of the structure, there are leaf-nodes which contains values (geometric objects
        representations).
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../img/index/rtree/rstar.png" alt="rstar"></span>
      </p>
<p>
        The R-tree is a self-balanced data structure. The key part of balancing algorithm
        is node splitting algorithm <a href="#ftn.geometry.spatial_indexes.introduction.f2" class="footnote" name="geometry.spatial_indexes.introduction.f2"><sup class="footnote">[3]</sup></a> <a href="#ftn.geometry.spatial_indexes.introduction.f3" class="footnote" name="geometry.spatial_indexes.introduction.f3"><sup class="footnote">[4]</sup></a>. Each algorithm produces different splits so the internal structure
        of a tree may be different for each one of them. In general, more complex
        algorithms analyses elements better and produces less overlapping nodes.
        In the searching process less nodes must be traversed in order to find desired
        objects. On the other hand more complex analysis takes more time. In general
        faster inserting will result in slower searching and vice versa. The performance
        of the R-tree depends on balancing algorithm, parameters and data inserted
        into the container.
      </p>
<p>
        Additionally there are also algorithms creating R-tree containing some, number
        of objects. This technique is called bulk loading and is done by use of packing
        algorithm <a href="#ftn.geometry.spatial_indexes.introduction.f4" class="footnote" name="geometry.spatial_indexes.introduction.f4"><sup class="footnote">[5]</sup></a> <a href="#ftn.geometry.spatial_indexes.introduction.f5" class="footnote" name="geometry.spatial_indexes.introduction.f5"><sup class="footnote">[6]</sup></a>. This method is faster and results in R-trees with better internal
        structure. This means that the query performance is increased.
      </p>
<p>
        The examples of structures of trees created by use of different algorithms
        and exemplary operations times are presented below.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  Linear algorithm
                </p>
              </th>
<th>
                <p>
                  Quadratic algorithm
                </p>
              </th>
<th>
                <p>
                  R*-tree
                </p>
              </th>
<th>
                <p>
                  Packing algorithm
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  <span class="bold"><strong>Example structure</strong></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/linear.png" alt="linear"></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/quadratic.png" alt="quadratic"></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/rstar.png" alt="rstar"></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/bulk.png" alt="bulk"></span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <span class="bold"><strong>1M Values inserts</strong></span>
                </p>
              </td>
<td>
                <p>
                  1.76s
                </p>
              </td>
<td>
                <p>
                  2.47s
                </p>
              </td>
<td>
                <p>
                  6.19s
                </p>
              </td>
<td>
                <p>
                  0.64s
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <span class="bold"><strong>100k spatial queries</strong></span>
                </p>
              </td>
<td>
                <p>
                  2.21s
                </p>
              </td>
<td>
                <p>
                  0.51s
                </p>
              </td>
<td>
                <p>
                  0.12s
                </p>
              </td>
<td>
                <p>
                  0.07s
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <span class="bold"><strong>100k knn queries</strong></span>
                </p>
              </td>
<td>
                <p>
                  6.37s
                </p>
              </td>
<td>
                <p>
                  2.09s
                </p>
              </td>
<td>
                <p>
                  0.64s
                </p>
              </td>
<td>
                <p>
                  0.52s
                </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        The configuration of the machine used for testing was: <span class="emphasis"><em>Intel(R)
        Core(TM) i7 870 @ 2.93GHz, 8GB RAM, MS Windows 7 x64</em></span>. The code
        was compiled with optimization for speed (<code class="computeroutput"><span class="identifier">O2</span></code>).
      </p>
<p>
        The performance of the R-tree for different values of Max parameter and Min=0.5*Max
        is presented in the table below. In the two upper figures you can see the
        performance of the R-tree storing random, relatively small, non-overlapping,
        2d boxes. In the lower ones, the performance of the R-tree also storing random,
        2d boxes, but this time quite big and possibly overlapping. As you can see,
        the R-tree performance is different in both cases.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  building
                </p>
              </th>
<th>
                <p>
                  querying
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  <span class="bold"><strong>non overlapping</strong></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/build_non_ovl.png" alt="build_non_ovl"></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/query_non_ovl.png" alt="query_non_ovl"></span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  <span class="bold"><strong>overlapping</strong></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/build_ovl.png" alt="build_ovl"></span>
                </p>
              </td>
<td>
                <p>
                  <span class="inlinemediaobject"><img src="../../img/index/rtree/query_ovl.png" alt="query_ovl"></span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="geometry.spatial_indexes.introduction.h1"></a>
        <span class="phrase"><a name="geometry.spatial_indexes.introduction.implementation_details"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.implementation_details">Implementation
        details</a>
      </h5>
<p>
        Key features of this implementation of the R-tree are:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            capable to store arbitrary Value type,
          </li>
<li class="listitem">
            three different balancing algorithms - linear, quadratic or rstar,
          </li>
<li class="listitem">
            creation using packing algorithm,
          </li>
<li class="listitem">
            parameters (including maximal and minimal number of elements) may be
            passed as compile- or run-time parameters, in compile-time version nodes
            elements are stored in static-size containers,
          </li>
<li class="listitem">
            advanced queries, e.g. search for 5 nearest Values to some point and
            intersecting some Geometry but not within the other one,
          </li>
<li class="listitem">
            iterative queries by use of iterators,
          </li>
<li class="listitem">
            C++11 conformant - move semantics, stateful allocators,
          </li>
<li class="listitem">
            capable to store Value type with no default constructor,
          </li>
<li class="listitem">
            in-memory storage by use of the default std::allocator&lt;&gt;,
          </li>
<li class="listitem">
            other storage options - shared memory and mapped file by use of Boost.Interprocess
            allocators.
          </li>
</ul></div>
<h5>
<a name="geometry.spatial_indexes.introduction.h2"></a>
        <span class="phrase"><a name="geometry.spatial_indexes.introduction.dependencies"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.dependencies">Dependencies</a>
      </h5>
<p>
        R-tree depends on Boost.Container, Boost.Core, Boost.Move, Boost.MPL, Boost.Range,
        Boost.Tuple.
      </p>
<h5>
<a name="geometry.spatial_indexes.introduction.h3"></a>
        <span class="phrase"><a name="geometry.spatial_indexes.introduction.contributors"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.contributors">Contributors</a>
      </h5>
<p>
        The spatial index was originally started by Federico J. Fernandez during
        the Google Summer of Code 2008 program, mentored by Hartmut Kaiser.
      </p>
<h5>
<a name="geometry.spatial_indexes.introduction.h4"></a>
        <span class="phrase"><a name="geometry.spatial_indexes.introduction.spatial_thanks"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.spatial_thanks">Spatial thanks</a>
      </h5>
<p>
        I'd like to thank Barend Gehrels, Bruno Lalande, Mateusz Łoskot, Lucanus
        J. Simonson for their support and ideas.
      </p>
<div class="footnotes">
<br><hr style="width:100; text-align:left;margin-left: 0">
<div id="ftn.geometry.spatial_indexes.introduction.f0" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f0" class="para"><sup class="para">[1] </sup></a>
          Guttman, A. (1984). <span class="emphasis"><em>R-Trees: A Dynamic Index Structure for Spatial
          Searching</em></span>
        </p></div>
<div id="ftn.geometry.spatial_indexes.introduction.f1" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f1" class="para"><sup class="para">[2] </sup></a>
          Cheung, K.; Fu, A. (1998). <span class="emphasis"><em>Enhanced Nearest Neighbour Search
          on the R-tree</em></span>
        </p></div>
<div id="ftn.geometry.spatial_indexes.introduction.f2" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f2" class="para"><sup class="para">[3] </sup></a>
          Greene, D. (1989). <span class="emphasis"><em>An implementation and performance analysis
          of spatial data access methods</em></span>
        </p></div>
<div id="ftn.geometry.spatial_indexes.introduction.f3" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f3" class="para"><sup class="para">[4] </sup></a>
          Beckmann, N.; Kriegel, H. P.; Schneider, R.; Seeger, B. (1990). <span class="emphasis"><em>The
          R*-tree: an efficient and robust access method for points and rectangles</em></span>
        </p></div>
<div id="ftn.geometry.spatial_indexes.introduction.f4" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f4" class="para"><sup class="para">[5] </sup></a>
          Leutenegger, Scott T.; Edgington, Jeffrey M.; Lopez, Mario A. (1997).
          <span class="emphasis"><em>STR: A Simple and Efficient Algorithm for R-Tree Packing</em></span>
        </p></div>
<div id="ftn.geometry.spatial_indexes.introduction.f5" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f5" class="para"><sup class="para">[6] </sup></a>
          Garcia, Yvan J.; Lopez, Mario A.; Leutenegger, Scott T. (1997). <span class="emphasis"><em>A
          Greedy Algorithm for Bulk Loading R-trees</em></span>
        </p></div>
</div>
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      Bruno Lalande, Mateusz Loskot, Adam Wulkiewicz, Oracle and/or its affiliates<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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